hyperelliptic(.ncag).info

the classification of hyperelliptic varieties in dimensions 2, 3 and 4

$\mathrm{C}_{6}$ in dimension 3

holonomy group
$\mathrm{C}_{6}$, order $6$, SmallGroup $[6,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, \zeta_{3})$
order of $\omega_X$
6
number of moduli
4
irregularity $q = \dim \operatorname{Aut}^0(X)$
2
Albanese
the Albanese variety has dimension $2$; the general fiber has dimension $1$ and is an elliptic curve (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 4, 6, 4, 1, 0, 0)$
Hodge diamond
1
2 2
1 5 1
0 4 4 0
1 5 1
2 2
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
22
141
0220
010
00
0

twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$

Since $\operatorname{ord} \omega_X = 6$, the twists by powers of the canonical bundle form a finite package of $6$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
22
151
0440
151
22
1
$\omega_X^{\otimes 1}$
0
01
022
0141
022
01
0
$\omega_X^{\otimes 2}$
0
00
000
0000
000
00
0
$\omega_X^{\otimes 3}$
0
00
000
0000
000
00
0
$\omega_X^{\otimes 4}$
0
00
000
0000
000
00
0
$\omega_X^{\otimes 5}$
0
10
220
1410
220
10
0

References

  1. K. Uchida, H. Yoshihara, Discontinuous groups of affine transformations of $\mathbb{C}^3$, Tôhoku Math. J. (2) 28 (1976) 89–94. MR400271 doi
  2. H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi
  3. F. Catanese, A. Demleitner, The classification of hyperelliptic threefolds, Groups Geom. Dyn. 14 (2020) 1447–1454. MR4186481 doi
  4. F. Catanese, A. Demleitner, Hyperelliptic threefolds with group $\mathrm{D}_4$. arXiv:1805.01835
  5. P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi